To my knowledge there has been very little done in the 21st century.
Here is a subsection of a page on space elevators and orbital tethers that lists tethers missions. The last two paragraphs are discouraging:
Tethers Unlimited flew a tether demonstration named the
"Multi-Application Survivable Tether (MAST)" experiment as part of a
set of a small spacecraft launched by a Russian Dnepr booster in 2007.
MAST was based on three "CubeSats", which are small standardized
satellites in the form of cubes 10 centimeters on a side, intended to
provide low-cost access to space. MAST consisted of a 1-kilometer-long
tether, made of three braided strands, with a CubeSat at each end and
a third CubeSat, named "Gadget", moving up and down the tether to
inspect it. All three satellites were fitted with GPS receivers for
tracking. The system failed to deploy properly.
Since MAST, several further space tether experiments have been
performed, including a number of CubeSats -- not one of which proved
any more successful than MAST, and which need not be further discussed
here. Activity on space tethers is effectively stalled. Hopefully,
things will improve in the future.
There is a proposed NASA mission to Phobos which would dangle a tether from the Mars Phobos L1 region down to just above Phobos' surface. I'm excited above this proposal as Martian moon tethers are some of my favorite day dreams. See the pieces I've written: Phobos, Panama Canal of the inner solar system, Upper Phobos tether and Deimos tether. PHLOTE might be a precursor to the infrastructure I like to imagine. However this is another project that still hasn't made it past the Power Point phase (at the time of this writing).
I was pleased to see a recent Kurzgesagt video on sky hooks. This Youtube channel has 10.2 million subscribers so it has generated more public interest in orbital tethers. By and large the video's pretty sound but it's led some people to ask "if sky hooks are so great, why aren't they being developed?" I will look at some of the obstacles.
Kurzgesagt correctly notes that, given two way traffic, an orbital tether can trade up momentum with down momentum and thus maintain its orbit with very little reaction mass (aka propellant). Which would indeed mitigate some of the difficult engineering challenges imposed by the rocket equation and large delta V budgets. However this presupposes two way traffic. At this time we're only sending stuff one way -- up. To trade momentum we would need to be importing mass from higher orbits. There are possible sources for this up momentum: propellent and other commodities mined from the moon or near earth asteroids. However at this time these sources of up momentum don't exist. Until they do every payload sent up will subtract from the tether's orbital momentum, a loss that must be restored with reaction mass.
Another obstacle is the need for a very substantial anchor mass. If the sky hook's mass is in the same ball park as the payload mass, a single catch would drag it down to the upper atmosphere. In Zubrin's HASTOL paper he suggests the sky hook mass exceed payload mass by a factor of 200. To accommodate a 500 kilogram communications satellite you would need a sky hook about twice the mass of the International Space Station.
After Musk had successfully landed a Booster on an ocean platform Carmack tweeted that he could manage a rotovator rendezvous. Musk agreed but with the caveat "Maybe as a future optimization. Would only matter if it was extremely big." I believe Musk was referring to a rotovator's need for a huge anchor mass.
Another obstacle is the need for trial and error engineering. Think of a tether as a giant fishing reel. When I was learning to fish I spent more time untangling my reel than fishing. And I'd frequently break my line. I believe tether engineers will face a similar learning curve. If you look at the Tarokt list of tether missions you will see a number of missions that failed to deploy. I suspect robotically reeling and unreeling lines is difficult.
Kurzgesagt seems to imagine a LEO rotovator throwing payloads into interplanetary orbits. This is one aspect of the video I think is unrealistic. Like the rocket equation, the Moravec and Pearson equations are exponential. Tether mass goes up exponentially as the delta V budget grows. A LEO rotovator capable of 3 or 4 km/s delta V is horribly impractical. Much more doable is a system of 3 tethers, each imparting 1 km/s delta V. The tethers would trade payloads via ZRVTOs (Zero Relative Velocity Transfer Orbits). For anchor mass the tether just above geosynchronous orbit might harvest dead sats in graveyard orbits just above geosynch. There are roughly 670 tonnes of dead sats in near equatorial orbits at this altitude.